Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane (for instance), can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. V i , y = ∑ j = 1 n ∂ y i ∂ x j V j , x . {displaystyle V_{i,y}=sum _{j=1}^{n}{frac {partial y_{i}}{partial x_{j}}}V_{j,x}.}     (1) In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane (for instance), can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow). In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field. Given a subset S in Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian coordinates (x1, ..., xn). If each component of V is continuous, then V is a continuous vector field, and more generally V is a Ck vector field if each component of V is k times continuously differentiable. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space. Given two Ck-vector fields V, W defined on S and a real valued Ck-function f defined on S, the two operations scalar multiplication and vector addition define the module of Ck-vector fields over the ring of Ck-functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being fid(p) := 1). In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.